Question

The equation $\sqrt{3}\sin x+\cos x=4$ has

• A. infinitely many solutions
• B. no solution
• C. two solutions
• D. only one solution

Answer 1

Answer: (B)

We know that,
$-\sqrt{a^2+b^2}\le a\cos \theta +b\sin \theta \le \sqrt{a^2+b^2}$
$\therefore -\sqrt{3+1}\le \sqrt{3}\sin x+\cos x\le \sqrt{3+1}$
$\Rightarrow -2\le \sqrt{3}\sin x+\cos x\le \sqrt{2}$
But $\sqrt{3}\sin x+\cos x=4$
Hence, given equation has no solution.